The bullet passing through the water creates a pressure wave. When the wave hits the side and bottom of the plastic water jug, the force of the wave is passed through the plastic wall of the jug and into whatever is on the other side. The bottom of the jug is resting on a firm solid surface, so the plastic flexible bottom "punches" the floor, causing the jug to jump.

If you could have your hand on the side of the jug as the bullet passed through, you would feel a sharp impact in your hand. I would imagine it would be similar to catching a baseball. I don't recommend this !

If the jug were sitting on an absorbent foam pad, it probably would not jump.

If the jug were a steel drum of the same weight, it probably would not jump because the steel would flex less and transmit less of the wave energy into the floor. In fact most steel drums are designed so that only the rim touches the floor, the bottom of the drum is raised above the floor. In this case, almost no energy would be transmitted to the floor.... no jump.